So far, we have defined a sequence of numbers , and showed that
where and . This is a big step: the are defined recursively (that is, each is defined in terms of the previous ), but and give us a direct arithmetic expression for , which can make proving things about the easier. So our next goal will be to better understand .
As a warmup, consider that and are both of the form , where and are integers. Consider the set of all such numbers, call it . So, for example, contains things like and and and … and so on. (Mathematicians would call this set , but it doesn’t really matter what we call it.) First of all, note that if we take any two numbers in and add them, we get another number in :
For example, . A shorter way to say this is that is “closed under addition”. The additive identity, , is also a member of , since .
We can show that is closed under multiplication, too:
And the multiplicative identity is likewise in , since .
So is a sort of “system of numbers” in a similar sense to the integers, or rational numbers, or especially the complex numbers . At this point we could start talking about rings and fields and stuff, but we don’t actually need anything that complicated. We just need to talk a little bit about groups. I’ve been wanting to write about groups on this blog for a while, and this is a great excuse! So, tomorrow, we’ll start to learn about groups and develop some of the ideas we will need to understand .